Date: 20/06/13 Location: Carslaw 535 Time: 15:05-16:00 Name: Mark Watkins Affiliation: University of Sydney Title: Ranks of congruent number twists Abstract: For each positive $d$, the elliptic curve $d*y^2 = x^3 - x$ has positive rank if and only if $d$ is the area of a right-angled triangle with rational sides, the latter being a question of antiquity. We do not address this question of positive rank per se, but rather present the results of some experiments trying to find $d$ such that the above elliptic curve has large rank. These can be seen as $d$ for which there exist "many" rational-sided triangles of area $d$. Date: 20/06/13 Location: Carslaw 535 Time: 16:15-17:10 Name: Brendan Creutz Affiliation: University of Sydney Title: 2-torsion Brauer classes on double covers. Abstract: Let X be a smooth double cover of a geometrically ruled surface over a separably closed field of characteristic different from 2. I will discuss recent work in which we give a finite presentation of the two-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Neron-Severi group of X. I will also discuss some of the motivation for this coming from arithmetic applications such as computing Brauer-Manin obstructions to the existence of rational points. This is joint work with Bianca Viray.